Question

Find the critical value(s) of \(t\) that specify the rejection region in these situations: a. A two-tailed test with \(\alpha=.01\) and \(12 d f\) b. A right-tailed test with \(\alpha=.05\) and \(16 d f\) c. A two-tailed test with \(\alpha=.05\) and \(25 d f\) d. A left-tailed test with \(\alpha=.01\) and \(7 d f\)

Short answer

#Short Answer# a. The critical values for a two-tailed test with α = 0.01 and 12 d.f. are approximately ±3.106. b. The critical value for a right-tailed test with α = 0.05 and 16 d.f. is approximately 1.746. c. The critical values for a two-tailed test with α = 0.05 and 25 d.f. are approximately ±2.060. d. The critical value for a left-tailed test with α = 0.01 and 7 d.f. is approximately -2.998.

Step-by-step solution

Test Type

This is a two-tailed test. We will use α/2 = 0.005 to find the critical values on each tail. ##Step 2: Identify the degrees of freedom##

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 12. ##Step 3: Find the critical value##

Critical Value

Using a t-distribution table or calculator, we find the critical values for α/2 = 0.005 and 12 d.f. The critical values are approximately ±3.106. ##Step 4: Conclusion##

Conclusion

The critical values for a two-tailed test with α = 0.01 and 12 d.f. are approximately ±3.106. #b. A right-tailed test with α = .05 and 16 df# ##Step 1: Determine the test type##

Test Type

This is a right-tailed test. We will use α = 0.05. ##Step 2: Identify the degrees of freedom##

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 16. ##Step 3: Find the critical value##

Critical Value

Using a t-distribution table or calculator, we find the critical value for α = 0.05 and 16 d.f. The critical value is approximately 1.746. ##Step 4: Conclusion##

Conclusion

The critical value for a right-tailed test with α = 0.05 and 16 d.f. is approximately 1.746. #c. A two-tailed test with α = .05 and 25 df# ##Step 1: Determine the test type and α/2##

Test Type

This is a two-tailed test. We will use α/2 = 0.025. ##Step 2: Identify the degrees of freedom##

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 25. ##Step 3: Find the critical value##

Critical Value

Using a t-distribution table or calculator, we find the critical values for α/2 = 0.025 and 25 d.f. The critical values are approximately ±2.060. ##Step 4: Conclusion##

Conclusion

The critical values for a two-tailed test with α = 0.05 and 25 d.f. are approximately ±2.060. #d. A left-tailed test with α = .01 and 7 df# ##Step 1: Determine the test type##

Test Type

This is a left-tailed test. We will use α = 0.01. ##Step 2: Identify the degrees of freedom##

Degrees of Freedom

In this case, degrees of freedom (d.f.) = 7. ##Step 3: Find the critical value##

Critical Value

Using a t-distribution table or calculator, we find the critical value for α = 0.01 and 7 d.f. The critical value is approximately -2.998. ##Step 4: Conclusion##

Conclusion

The critical value for a left-tailed test with α = 0.01 and 7 d.f. is approximately -2.998.

Key concepts

Critical Value

The critical value in a t-distribution is an essential component in statistical hypothesis testing. It helps to determine whether the null hypothesis should be rejected. The critical value marks the boundaries of the rejection region, which is the area in the tails of the distribution that lies beyond a certain chance of error, known as the level of significance (\( \alpha \)).

• For a two-tailed test, you need to divide the level of significance by two since the rejection area is split between both tails of the distribution.
• For one-tailed tests, the entire significance level is in one tail.
• Finding a critical value requires using a t-distribution table, or a calculator, based on degrees of freedom and the chosen significance level.

These critical values help decide whether the test statistic falls within the critical region, leading to the rejection of the null hypothesis.

Degrees of Freedom

Degrees of freedom (\( \text{d.f.} \)) are a crucial aspect of t-distributions and greatly impact the shape of the distribution. It essentially refers to the number of independent observations in a sample minus the number of parameters estimated. The calculation of degrees of freedom in most t-tests is straightforward:

• For a single sample, the degrees of freedom are equal to the sample size minus one (\( n - 1 \)).
• More complex designs, such as paired samples or multiple groups, may involve more intricate calculations.

Degrees of freedom affect the spread and the tail areas of the t-distribution. Lower degrees of freedom result in fatter tails, indicating more variability. As the degrees of freedom increase, the t-distribution approaches a normal distribution.

Two-Tailed Test

A two-tailed test is a statistical test where the area of rejection is on both ends of the sampling distribution. This type of test is used when we are interested in determining whether there is a difference regardless of direction.

• In hypothesis testing, it looks for deviations in either direction from the hypothesized parameter.
• For example, if testing whether a sample mean is different from a known population mean, you would check for values significantly higher or lower.

The significance level for a two-tailed test is divided between the two tails. So if \( \alpha = 0.05 \), then each tail would have a 0.025 level of significance. This approach is more conservative, as it requires stronger evidence to reject the null hypothesis since it checks both directions.

Right-Tailed Test

In a right-tailed test, the rejection region is in the right tail of the distribution. This type of test is specifically used when predicting an increase or greater values in the parameter of interest.

• For example, if testing whether a new drug increases response times, a right-tailed test could be appropriate.
• The critical value separates the non-rejection area from the rejection region in the right tail.

The entire chosen significance level (\( \alpha \)) is in the right side tail. Thus, for \( \alpha = 0.05 \), 5% of the distribution area falls under the rejection region. Right-tailed tests are pivotal in hypotheses where you only care about deviations in one direction, specifically to the right.

Left-Tailed Test

A left-tailed test, meanwhile, sets its sights on the left end of the distribution. It's most applicable when a decrease or lesser value in the parameter in question is the point of interest.

• For example, if assessing whether a new teaching technique lowers anxiety levels, a left-tailed test could be used.
• The critical value defines the boundary beyond which the null hypothesis will be rejected.

Here, the entire significance level (\( \alpha \)) resides in the left tail. For instance, with a significance level of 0.01, 1% of the area on the left tail makes up the rejection region. The left-tailed test is perfect for scenarios where changes suspected are directionally specific to the left.